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      })(() => window.markmap,null,{"t":"heading","d":1,"p":{"lines":[0,1]},"v":"CRF","c":[{"t":"heading","d":2,"p":{"lines":[2,3]},"v":"概念引入","c":[{"t":"heading","d":3,"p":{"lines":[3,4]},"v":"联合概率分布","c":[{"t":"list_item","d":5,"p":{"lines":[4,6]},"v":"是两个及以上随机变量组成的随机向量的概率分布，<br>\n指的是包含多个条件且所有条件同时成立的概率。"}]}]},{"t":"heading","d":2,"p":{"lines":[7,8]},"v":"图","c":[{"t":"heading","d":3,"p":{"lines":[8,9]},"v":"图定义","c":[{"t":"list_item","d":5,"p":{"lines":[9,12]},"v":"图是由结点及连接结点的边组成的集合<br>\n结点和边分别记作<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>e</mi></mrow><annotation encoding=\"application/x-tex\">e</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">e</span></span></span></span>，结点和边的集合分别记作<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>V</mi></mrow><annotation encoding=\"application/x-tex\">V</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">V</span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>E</mi></mrow><annotation encoding=\"application/x-tex\">E</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span></span></span></span><br>\n图记作<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mi>V</mi><mo separator=\"true\">,</mo><mi>E</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">G=(V,E)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">V</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mclose\">)</span></span></span></span>"}]},{"t":"heading","d":3,"p":{"lines":[12,13]},"v":"无向图","c":[{"t":"list_item","d":5,"p":{"lines":[13,14]},"v":"指边没有方向的图"}]},{"t":"heading","d":3,"p":{"lines":[14,15]},"v":"概率图模型","c":[{"t":"list_item","d":5,"p":{"lines":[15,19]},"v":"概率图模型是由图表示的概率分布。<br>\n无向图<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy=\"false\">(</mo><mi>V</mi><mo separator=\"true\">,</mo><mi>E</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">G=(V,E)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">V</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mclose\">)</span></span></span></span>表示概率分布<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span></span></span></span>，<br>\n节点<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding=\"application/x-tex\">v\\in V</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">V</span></span></span></span>表示一个随机变量<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>V</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_V</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">V</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>；<br>\n边<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding=\"application/x-tex\">e\\in E</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">e</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span></span></span></span>表示随机变量之间的概率依存关系。"},{"t":"list_item","d":5,"p":{"lines":[19,21]},"v":"给定一个联合概率分布<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span></span></span></span>和表示它的无向图<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi></mrow><annotation encoding=\"application/x-tex\">G</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span></span></span></span>。<br>\n定义无向图表示的随机变量之间存在马尔可夫性。"}]},{"t":"heading","d":3,"p":{"lines":[23,24]},"v":"马尔可夫性","c":[{"t":"heading","d":4,"p":{"lines":[24,25]},"v":"成对马尔科夫性","c":[{"t":"list_item","d":6,"p":{"lines":[25,27]},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>u</mi></mrow><annotation encoding=\"application/x-tex\">u</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">u</span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>是<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi></mrow><annotation encoding=\"application/x-tex\">G</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span></span></span></span>中任意两个没有边连接的节点，其他所有节点为<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>O</mi></mrow><annotation encoding=\"application/x-tex\">O</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">O</span></span></span></span>。<br>\n成对马尔科夫性是指给定随机变量组<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>O</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_O</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">O</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>的条件下随机变量<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>u</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_u</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>v</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>是条件独立的，即"},{"t":"list_item","d":6,"p":{"lines":[27,28]},"v":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>u</mi></msub><mo separator=\"true\">,</mo><msub><mi>Y</mi><mi>v</mi></msub><mi mathvariant=\"normal\">∣</mi><msub><mi>Y</mi><mi>O</mi></msub><mo stretchy=\"false\">)</mo><mo>=</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>u</mi></msub><mi mathvariant=\"normal\">∣</mi><msub><mi>Y</mi><mi>O</mi></msub><mo stretchy=\"false\">)</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>v</mi></msub><mi mathvariant=\"normal\">∣</mi><msub><mi>Y</mi><mi>O</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y_u,Y_v|Y_O)=P(Y_u|Y_O)P(Y_v|Y_O)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">O</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">u</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">O</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">O</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>"}]},{"t":"heading","d":4,"p":{"lines":[28,29]},"v":"局部马尔科夫性","c":[{"t":"list_item","d":6,"p":{"lines":[29,31]},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>是<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi></mrow><annotation encoding=\"application/x-tex\">G</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span></span></span></span>中任意一点，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>W</mi></mrow><annotation encoding=\"application/x-tex\">W</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span></span></span></span>是与<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>有边连接的所有节点，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>O</mi></mrow><annotation encoding=\"application/x-tex\">O</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">O</span></span></span></span>是<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>W</mi></mrow><annotation encoding=\"application/x-tex\">W</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">W</span></span></span></span>以外的其他所有节点。<br>\n局部马尔科夫性是指给定<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>W</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_W</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">W</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>的条件下<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>v</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>与<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>O</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_O</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">O</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>是独立的，即"},{"t":"list_item","d":6,"p":{"lines":[31,32]},"v":"P(Y_v,Y_O|Y_W)=P(Y_v|Y_W)P(Y_O|Y_W)"}]},{"t":"heading","d":4,"p":{"lines":[32,33]},"v":"全局马尔科夫性","c":[{"t":"list_item","d":6,"p":{"lines":[33,35]},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>A</mi><mo separator=\"true\">,</mo><mi>B</mi></mrow><annotation encoding=\"application/x-tex\">A,B</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\">A</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05017em;\">B</span></span></span></span>是<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi></mrow><annotation encoding=\"application/x-tex\">G</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span></span></span></span>中被<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>C</mi></mrow><annotation encoding=\"application/x-tex\">C</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span></span></span></span>分开的任意节点集合。<br>\n全局马尔科夫性是指给定<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>C</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_C</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>条件下<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>A</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_A</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">A</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>Y</mi><mi>B</mi></msub></mrow><annotation encoding=\"application/x-tex\">Y_B</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>是条件独立的，即"},{"t":"list_item","d":6,"p":{"lines":[35,36]},"v":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>A</mi></msub><mo separator=\"true\">,</mo><msub><mi>Y</mi><mi>B</mi></msub><mi mathvariant=\"normal\">∣</mi><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo><mo>=</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>A</mi></msub><mi mathvariant=\"normal\">∣</mi><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>B</mi></msub><mi mathvariant=\"normal\">∣</mi><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y_A,Y_B|Y_C)=P(Y_A|Y_C)P(Y_B|Y_C)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">A</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">A</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\">B</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>"}]},{"t":"heading","d":4,"p":{"lines":[36,37]},"v":"上述成对的、局部的、全局的马尔可夫性定义是等价的。"}]},{"t":"heading","d":3,"p":{"lines":[39,40]},"v":"概率无向图模型","c":[{"t":"heading","d":4,"p":{"lines":[40,41]},"v":"定义","c":[{"t":"list_item","d":6,"p":{"lines":[41,45]},"v":"如果联合概率<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span></span></span></span>满足成对、局部或者全局马尔科夫性，就称该联合概率分布为<strong>概率无向图模型</strong>，或者<strong>马尔科夫随机场</strong>。<br>\n这是概率无向图模型的定义，实际上，我们更关心的是如何求其联合概率分布。<br>\n对给定的概率无向图模型，我们希望将整体的联合概率写成若干子联合概率的乘积形式，也就是将联合概率进行因子分解，这样便于模型的学习与计算。<br>\n事实上，概率无向图模型最大特点：易于因子分解。"}]},{"t":"heading","d":4,"p":{"lines":[45,46]},"v":"团与最大团","c":[{"t":"list_item","d":6,"p":{"lines":[46,47]},"v":"无向图<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi></mrow><annotation encoding=\"application/x-tex\">G</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span></span></span></span>中任何两个节点都有边连接的节点子集称为团（clique）。若不能再加进一个节点使团更大，称该团为最大团。"}]},{"t":"heading","d":4,"p":{"lines":[47,48]},"v":"无向图模型的因子分解","c":[{"t":"list_item","d":6,"p":{"lines":[48,49]},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>C</mi></mrow><annotation encoding=\"application/x-tex\">C</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span></span></span></span>为<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi></mrow><annotation encoding=\"application/x-tex\">G</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\">G</span></span></span></span>上最大团，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span></span></span></span>可以写作图中所有最大团<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>C</mi></mrow><annotation encoding=\"application/x-tex\">C</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span></span></span></span>上的函数<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi mathvariant=\"normal\">Ψ</mi><mi>c</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\Psi_c(Y_C)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\">Ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">c</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>的乘积形式，即"},{"t":"list_item","d":6,"p":{"lines":[49,50]},"v":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>Z</mi></mfrac><munder><mo>∏</mo><mi>C</mi></munder><msub><mi mathvariant=\"normal\">Ψ</mi><mi>C</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y)=\\frac{1}{Z} \\prod_{C}\\Psi_C(Y_C)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.6158em;vertical-align:-1.2943em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">Z</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8557em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∏</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2943em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\">Ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>"},{"t":"list_item","d":6,"p":{"lines":[50,51]},"v":"其中Z是归一化因子，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>Z</mi><mo>=</mo><msub><mo>∑</mo><mi>Y</mi></msub><msub><mo>∏</mo><mi>C</mi></msub><msub><mi mathvariant=\"normal\">Ψ</mi><mi>C</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">Z=\\sum_Y\\prod_C\\Psi_C(Y_C)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">Z</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0497em;vertical-align:-0.2997em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1786em;\"><span style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">Y</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2997em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∏</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1786em;\"><span style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2997em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\">Ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi mathvariant=\"normal\">Ψ</mi><mi>C</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\Psi_C(Y_C)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\">Ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>称为势函数，通常定义为<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi mathvariant=\"normal\">Ψ</mi><mi>C</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo><mo>=</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy=\"false\">{</mo><mo>−</mo><mi>E</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">}</mo></mrow><annotation encoding=\"application/x-tex\">\\Psi_C(Y_C)=exp\\{-E(Y_C)\\}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\">Ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">e</span><span class=\"mord mathnormal\">x</span><span class=\"mord mathnormal\">p</span><span class=\"mopen\">{</span><span class=\"mord\">−</span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)}</span></span></span></span>"}]},{"t":"heading","d":4,"p":{"lines":[51,52]},"v":"概率无向图的因子分解由下述定理保证","c":[{"t":"list_item","d":6,"p":{"lines":[52,54]},"v":"Hammersley-Clifford定理<br>\n概率无向图模型的联合概率分布P(Y)可以表示为："},{"t":"list_item","d":6,"p":{"lines":[54,56]},"v":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>Z</mi></mfrac><munder><mo>∏</mo><mi>C</mi></munder><msub><mi mathvariant=\"normal\">Ψ</mi><mi>C</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo><mspace linebreak=\"newline\"></mspace><mi>Z</mi><mo>=</mo><munder><mo>∑</mo><mi>Y</mi></munder><munder><mo>∏</mo><mi>C</mi></munder><msub><mi mathvariant=\"normal\">Ψ</mi><mi>C</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>C</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y)=\\frac{1}{Z} \\prod_{C}\\Psi_C(Y_C)\\\\ Z=\\sum_Y\\prod_C\\Psi_C(Y_C)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.6158em;vertical-align:-1.2943em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">Z</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8557em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∏</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2943em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\">Ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span><span class=\"mspace newline\"></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">Z</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:2.3443em;vertical-align:-1.2943em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8557em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">Y</span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2943em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8557em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∏</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.2943em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord\">Ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>"}]}]}]},{"t":"heading","d":2,"p":{"lines":[57,58]},"v":"条件随机场","c":[{"t":"heading","d":3,"p":{"lines":[58,59]},"v":"定义","c":[{"t":"list_item","d":5,"p":{"lines":[59,61]},"v":"条件随机场是给定随机变量<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>X</mi></mrow><annotation encoding=\"application/x-tex\">X</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span></span></span></span>的条件下，随机变量Y的马尔科夫随机场。<br>\n主要介绍定义在线性链想的线性链条件随机场（用于标注等问题）"},{"t":"list_item","d":5,"p":{"lines":[61,62]},"v":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>v</mi></msub><mi mathvariant=\"normal\">∣</mi><mi>X</mi><mo separator=\"true\">,</mo><msub><mi>Y</mi><mi>w</mi></msub><mo separator=\"true\">,</mo><mi>w</mi><mo mathvariant=\"normal\">≠</mo><mi>v</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>v</mi></msub><mi mathvariant=\"normal\">∣</mi><mi>X</mi><mo separator=\"true\">,</mo><msub><mi>Y</mi><mi>w</mi></msub><mo separator=\"true\">,</mo><mi>w</mi><mtext>连</mtext><mi>v</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y_v|X, Y_w, w\\ne v) = P(Y_v|X, Y_w, w连v)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\"></span></span></span><span class=\"fix\"></span></span></span></span></span><span class=\"mrel\">=</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mord cjk_fallback\">连</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span><span class=\"mclose\">)</span></span></span></span></span>"},{"t":"list_item","d":5,"p":{"lines":[62,65]},"v":"对于任何节点<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>都成立，成条件概率分布<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mi mathvariant=\"normal\">∣</mi><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y|X)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mclose\">)</span></span></span></span>为条件随机场。<br>\n<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi></mrow><annotation encoding=\"application/x-tex\">w</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span></span></span></span>连<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>表示所有与<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>相连的节点<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi></mrow><annotation encoding=\"application/x-tex\">w</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span></span></span></span>，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi><mo mathvariant=\"normal\">≠</mo><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">w≠v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"inner\"><span class=\"mord\"><span class=\"mrel\"></span></span></span><span class=\"fix\"></span></span></span></span></span><span class=\"mrel\">=</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>表示所有除<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>外的节点<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi></mrow><annotation encoding=\"application/x-tex\">w</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span></span></span></span>。<br>\n也就是说，对于点<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi></mrow><annotation encoding=\"application/x-tex\">v</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span></span></span></span>来说，只有与它相连的点会对它产生影响。"}]},{"t":"heading","d":3,"p":{"lines":[66,67]},"v":"线性链条件随机场","c":[{"t":"list_item","d":5,"p":{"lines":[67,69]},"v":"设<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><mo>⋯</mo><mtext> </mtext><mo separator=\"true\">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><mi>Y</mi><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>Y</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><mo>⋯</mo><mtext> </mtext><mo separator=\"true\">,</mo><mi>Y</mi><mi>n</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">X=(X_1,X_2,\\cdots,X_n),Y=(Y_1,Y_2,\\cdots,Yn)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\">⋯</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\">⋯</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">Yn</span><span class=\"mclose\">)</span></span></span></span>均为线性链表示的随机变量序列，<br>\n若在给定随机变量序列<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>X</mi></mrow><annotation encoding=\"application/x-tex\">X</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span></span></span></span>的条件下，随机变量序列<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>Y</mi></mrow><annotation encoding=\"application/x-tex\">Y</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span></span></span></span>的条件概率分布<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>Y</mi><mi mathvariant=\"normal\">∣</mi><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y|X)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mclose\">)</span></span></span></span>构成条件随机场，即满足马尔可夫性"},{"t":"list_item","d":5,"p":{"lines":[69,71]},"v":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>i</mi></msub><mi mathvariant=\"normal\">∣</mi><mi>X</mi><mo separator=\"true\">,</mo><msub><mi>Y</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><mo>⋯</mo><mtext> </mtext><mo separator=\"true\">,</mo><msub><mi>Y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator=\"true\">,</mo><msub><mi>Y</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo separator=\"true\">,</mo><mo>⋯</mo><mtext> </mtext><mo separator=\"true\">,</mo><msub><mi>Y</mi><mi>n</mi></msub><mo stretchy=\"false\">)</mo><mo>=</mo><mi>P</mi><mo stretchy=\"false\">(</mo><msub><mi>Y</mi><mi>i</mi></msub><mi mathvariant=\"normal\">∣</mi><mi>X</mi><mo separator=\"true\">,</mo><msub><mi>Y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator=\"true\">,</mo><msub><mi>Y</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy=\"false\">)</mo><mspace linebreak=\"newline\"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo separator=\"true\">,</mo><mn>2</mn><mo separator=\"true\">,</mo><mo>⋯</mo><mtext> </mtext><mo separator=\"true\">,</mo><mi>n</mi><mtext>  </mtext><mtext>  </mtext><mo stretchy=\"false\">(</mo><mtext>在</mtext><mi>i</mi><mo>=</mo><mn>1</mn><mtext>和</mtext><mi>n</mi><mtext>时只考虑单边</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(Y_i|X,Y_1,\\cdots,Y_{i-1},Y_{i+1},\\cdots,Y_n)=P(Y_i|X,Y_{i-1},Y_{i+1}) \\\\ i=1,2,\\cdots,n\\;\\;(在i=1和n时只考虑单边)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\">⋯</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mbin mtight\">+</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\">⋯</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mbin mtight\">+</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span><span class=\"mspace newline\"></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6595em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">2</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\">⋯</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">n</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mopen\">(</span><span class=\"mord cjk_fallback\">在</span><span class=\"mord mathnormal\">i</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">1</span><span class=\"mord cjk_fallback\">和</span><span class=\"mord mathnormal\">n</span><span class=\"mord cjk_fallback\">时只考虑单边</span><span class=\"mclose\">)</span></span></span></span></span>"},{"t":"list_item","d":5,"p":{"lines":[71,73]},"v":"则称<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mi mathvariant=\"normal\">∣</mi><mi>Y</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(X|Y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span></span></span></span>为线性链条件随机场。在标注问题中，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>X</mi></mrow><annotation encoding=\"application/x-tex\">X</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span></span></span></span>表示输入观测序列，<br>\n<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>Y</mi></mrow><annotation encoding=\"application/x-tex\">Y</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span></span></span></span>表示对应的输出标记序列或者状态序列。一般将线性链条件随机场简称为条件随机场。"}]},{"t":"heading","d":3,"p":{"lines":[74,75]},"v":"条件随机场参数化形式","c":[{"t":"list_item","d":5,"p":{"lines":[75,76]},"v":"设<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mi mathvariant=\"normal\">∣</mi><mi>Y</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P(X|Y)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span><span class=\"mclose\">)</span></span></span></span>为线性链条件随机场，则在随机变量<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>X</mi></mrow><annotation encoding=\"application/x-tex\">X</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span></span></span></span>取值为<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi></mrow><annotation encoding=\"application/x-tex\">x</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\">x</span></span></span></span>的条件下，随机变量<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>Y</mi></mrow><annotation encoding=\"application/x-tex\">Y</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6833em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">Y</span></span></span></span>取值为<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>y</mi></mrow><annotation encoding=\"application/x-tex\">y</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span></span></span></span>的条件概率具有如下形式："},{"t":"list_item","d":5,"p":{"lines":[76,78]},"v":"<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mi mathvariant=\"normal\">∣</mi><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>Z</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow></mfrac><mi>exp</mi><mo>⁡</mo><mrow><mo fence=\"true\">(</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>k</mi></mrow></munder><msub><mi>λ</mi><mi>k</mi></msub><msub><mi>t</mi><mi>k</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo separator=\"true\">,</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>i</mi><mo stretchy=\"false\">)</mo><mo>+</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>l</mi></mrow></munder><msub><mi>μ</mi><mi>l</mi></msub><msub><mi>s</mi><mi>l</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo separator=\"true\">,</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>i</mi><mo stretchy=\"false\">)</mo><mo fence=\"true\">)</mo></mrow><mspace linebreak=\"newline\"></mspace><mtext>其中</mtext><mtext>  </mtext><mtext>  </mtext><mtext>  </mtext><mtext>  </mtext><mi>Z</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><munder><mo>∑</mo><mi>y</mi></munder><mi>exp</mi><mo>⁡</mo><mrow><mo fence=\"true\">(</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>k</mi></mrow></munder><msub><mi>λ</mi><mi>k</mi></msub><msub><mi>t</mi><mi>k</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo separator=\"true\">,</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>i</mi><mo stretchy=\"false\">)</mo><mo>+</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo separator=\"true\">,</mo><mi>l</mi></mrow></munder><msub><mi>μ</mi><mi>l</mi></msub><msub><mi>s</mi><mi>l</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo separator=\"true\">,</mo><mi>x</mi><mo separator=\"true\">,</mo><mi>i</mi><mo stretchy=\"false\">)</mo><mo fence=\"true\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">P(y|x)=\\frac{1}{Z(x)}\\exp\\left(\\sum_{i,k}\\lambda_k t_k (y_{i-1}, y_i, x, i)+\\sum_{i,l} \\mu_l s_l (y_i, x, i)\\right)\\\\ 其中\\;\\;\\;\\;Z(x)=\\sum_{y}\\exp\\left(\\sum_{i,k}\\lambda_k t_k (y_{i-1}, y_i, x, i)+\\sum_{i,l} \\mu_l s_l (y_i, x, i)\\right)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.6em;vertical-align:-1.55em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3214em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">Z</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.936em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">exp</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-2.25em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎝</span></span></span><span style=\"top:-3.397em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span style=\"height:0.016em;width:0.875em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M291 0 H417 V16 H291z M291 0 H417 V16 H291z'/></svg></span></span><span style=\"top:-4.05em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎛</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8479em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4382em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">λ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">t</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8479em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4382em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">s</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-2.25em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎠</span></span></span><span style=\"top:-3.397em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span style=\"height:0.016em;width:0.875em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M457 0 H583 V16 H457z M457 0 H583 V16 H457z'/></svg></span></span><span style=\"top:-4.05em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎞</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span></span></span></span><span class=\"mspace newline\"></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord cjk_fallback\">其中</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">Z</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:3.6em;vertical-align:-1.55em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.9em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.3861em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mop\">exp</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"minner\"><span class=\"mopen\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-2.25em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎝</span></span></span><span style=\"top:-3.397em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span style=\"height:0.016em;width:0.875em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M291 0 H417 V16 H291z M291 0 H417 V16 H291z'/></svg></span></span><span style=\"top:-4.05em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎛</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8479em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4382em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">λ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">t</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2083em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.05em;\"><span style=\"top:-1.8479em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span style=\"top:-3.05em;\"><span class=\"pstrut\" style=\"height:3.05em;\"></span><span><span class=\"mop op-symbol large-op\">∑</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.4382em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">s</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"delimsizing mult\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:2.05em;\"><span style=\"top:-2.25em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎠</span></span></span><span style=\"top:-3.397em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span style=\"height:0.016em;width:0.875em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M457 0 H583 V16 H457z M457 0 H583 V16 H457z'/></svg></span></span><span style=\"top:-4.05em;\"><span class=\"pstrut\" style=\"height:3.155em;\"></span><span class=\"delimsizinginner delim-size4\"><span>⎞</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.55em;\"><span></span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":5,"p":{"lines":[78,80]},"v":"式中，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>t</mi><mi>k</mi></msub></mrow><annotation encoding=\"application/x-tex\">t_k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7651em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">t</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>s</mi><mi>l</mi></msub></mrow><annotation encoding=\"application/x-tex\">s_l</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">s</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>是特征函数，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>λ</mi><mi>k</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\lambda_k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">λ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>μ</mi><mi>l</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\mu_l</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>是对应的权值。<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>Z</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">Z(x)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">Z</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span></span></span></span>是规范化因子，求和是在所有可能输出序列上进行的。<br>\n条件随机场完全由特征函数<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>t</mi><mi>k</mi></msub></mrow><annotation encoding=\"application/x-tex\">t_k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7651em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">t</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>s</mi><mi>l</mi></msub></mrow><annotation encoding=\"application/x-tex\">s_l</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">s</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>及其对应的权重<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>λ</mi><mi>k</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\lambda_k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">λ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>和<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>μ</mi><mi>l</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\mu_l</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>确定。"}]}]}]})</script>
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